Showing papers on "Prim's algorithm published in 2002"

An optimal minimum spanning tree algorithm

Abstract: We establish that the algorithmic complexity of the minimum spanning tree problem is equal to its decision-tree complexity. Specifically, we present a deterministic algorithm to find a minimum spanning tree of a graph with n vertices and m edges that runs in time O(T*(m,n)) where T* is the minimum number of edge-weight comparisons needed to determine the solution. The algorithm is quite simple and can be implemented on a pointer machine.Although our time bound is optimal, the exact function describing it is not known at present. The current best bounds known for T* are T*(m,n) = Ω(m) and T*(m,n) = O(m ∙ α(m,n)), where α is a certain natural inverse of Ackermann's function.Even under the assumption that T* is superlinear, we show that if the input graph is selected from Gn,m, our algorithm runs in linear time with high probability, regardless of n, m, or the permutation of edge weights. The analysis uses a new martingale for Gn,m similar to the edge-exposure martingale for Gn,p.

On the Implementation of MST-Based Heuristics for the Steiner Problem in Graphs

Abstract: Some of the most widely used constructive heuristics for the Steiner Problem in Graphs are based on algorithms for the Minimum Spanning Tree problem. In this paper, we examine efficient implementations of heuristics based on the classic algorithms by Prim, Kruskal, and Bor?vka. An extensive experimental study indicates that the theoretical worst-case complexity of the algorithms give little information about their behavior in practice. Careful implementation improves average computation times not only significantly, but asymptotically. Running times for our implementations are within a small constant factor from that of Prim's algorithm for the Minimum Spanning Tree problem, suggesting that there is little room for improvement.

A self-stabilizing algorithm for the Steiner tree problem

Abstract: Self-stabilization is a theoretical framework of non-masking fault-tolerant distributed algorithms. In this paper, we investigate the Steiner tree problem in distributed systems, and propose a self-stabilizing solution to the problem. Our solution is based on the pruned-MST technique, a heuristic technique to find a minimal cost Steiner tree by pruning unnecessary nodes and edges in a minimum cost spanning tree, provided that a minimum spanning tree is available. Finally we propose an algorithm to reduce the cost of the solution.

Optimality of the Tree Building Control Protocol

Abstract: This text focuses on the simulation and evaluation of Tree Building Control Protocol (TBCP) described in [1]. The TBCP is distributed algorithm used for building a spanning tree over TBCP entities. In this paper, the main part of the TBCP algorithm called join procedure is explained and the state machine of the join procedure is presented. Then, the non-distributed Branch and Bound algorithm finding an optimal spanning tree is shown. Finally, the TBCP is simulated in OPNET Modeler and compared to the optimal spanning tree.

Parallel algorithms for the degree-constrained minimum spanning tree problem using nearest-neighbor chains and the heap-traversal technique

Abstract: The degree-constrained minimum spanning tree (d-MST) problem attempts to find a minimum spanning tree with an added constraint that no nodes in the tree have a degree larger than a specified integer d. It is known that computing the d-MST is NP-hard for every d in the range 2 /spl les/ d /spl les/ (n - 2), where n denotes the total number of nodes. Several approximate algorithms (heuristics) have been proposed in the literature. We have previously proposed three approximate algorithms, IR, TC-RNN, and TC-NNC, for solving the d-MST problem, the last two (TC-RNN and TC-NNC) take advantage of nearest neighbors and their properties. Our experimental results showed that both the TC-RNN and TC-NNC algorithms consistently produce spanning trees with a smaller weight (better quality-of-solution) than that of IR, but using slightly longer execution time. We propose a new heap traversal technique that further improves the time efficiency of TC-RNN and TC-NNC. Our experiments using randomly generated, weighted graphs as inputs show that the TC-NNC algorithm outperforms the other two approximate algorithms in terms of the execution time and quality-of-solution.

Apparatus and method for accessing vertex data

Abstract: A vertex data access apparatus and method. The apparatus receives a vertex index, compares the vertex index with any vertices' indices used before, issues a request if necessary for fetching vertex data in system memory, stores the return vertex data in a vertex data queue and gets corresponding vertex data from the vertex data queue for further processing and, more particularly, if the vertex index is the same as one of those vertices' indices, the corresponding vertex data can be directly fetched from the vertex data queue. The vertex data queue performs the vertex cache function.

Associative parallel algorithms for computing functions defined on paths in trees

Abstract: By means of an abstract model of the SIMD type with vertical data processing (the STAR-machine), we present a simple associative parallel algorithm for finding tree paths in undirected graphs. We study applications of this algorithm to update minimum spanning trees in undirected graphs, to determine maximum flow values in a multiterminal network, and to find a fundamental set of circuits with respect to a given spanning tree. These algorithms are given as the corresponding STAR procedures whose correctness is proved and time complexity is evaluated.

Constant-time Algorithms for Minimum Spanning Tree and Related Problems on Processor Array with Reconfigurable Bus Systems

Abstract: A processor array with a reconfigurable bus system is a parallel computation model that consists of a processor array and a reconfigurable bus system. In this paper, a constant-time algorithm is proposed on this model for finding the cycles in an undirected graph. We can use this algorithm to decide whether a specified edge belongs to the minimum spanning tree of the graph or not. This cycle-finding algorithm is designed on a two-dimensional n×n processor array with a reconfigurable bus system, where n is the number of vertices in the graph. Based on this cycle-finding algorithm, the minimum spanning tree problem and the spanning tree problem can be solved in O(1) time by using fewer processors than before, O(n × m × n) and O(n3) processors respectively. This is a substantial improvement over previous known results. Moreover, we also propose two constanttime algorithms for solving the minimum spanning tree verification problem and spanning tree verification problem by using O(n3) and O(n2) processors, respectively.

Worst case analysis of a greedy multicast algorithm in k-ary n-cubes

Abstract: In this paper, we consider the problem of multicasting a message in k-ary n-cubes under the store-and-forward model. The objective of the problem is to minimize the size of the resultant multicast tree by keeping the distance to each destination over the tree the same as the distance in the original graph. In the following, we first propose an algorithm that grows a multicast tree in a greedy manner, in the sense that for each intermediate vertex of the tree, the outgoing edges of the vertex are selected in a non-increasing order of the number of destinations that can use the edge in a shortest path to the destination. We then evaluate the goodness of the algorithm in terms of the worst case ratio of the size of the generated tree to the size of an optimal tree. It is proved that for any k/spl ges/5 and n/spl ges/6, the performance ratio of the greedy algorithm is c/spl times/kn-o(n) for some constant 1/1.2/spl les/c/spl les/1/2.

Minimum Spanning Tree Algorithms Based on Genetic Algorithms

Abstract: Based on the graphic theory and genetic algorithm ,an improved genetic algorithm is introduced to search the minimum spanning trees This algorithm uses binary code to represent the problem of minimum spanning trees and uses the depth first searching method to determine the connectivity of the graphThe corresponding fitness function, single parent transposition operator, single parent reverse operators and four kinds of controlling evolutionary strategies are designed to improve its speed and efficiencyIn comparison with kruskal algorithm, it can acquire a set of minimum spanning trees during one genetic evolutionary process and is applicable to solving different kinds of minimum spanning tree problems